arXiv subjects
Publications and source records attributed to Juncheng Wei.
We study the Gauss curvature equation with negative singularities. For a local mean field type equation with only one negative index we prove a uniqueness property. For a global equation with one or two negative indexes we prove the non-degeneracy of the linearized equations.
We construct a smooth axially symmetric solution to the classical one phase free boundary problem in $\mathbb{R}^{N}$. Its free boundary is of \textquotedblleft catenoid\textquotedblright\ type. This is a higher dimensional analogy of the Hauswirth-Helein-Pacard solution in $\mathbb{R}% ^{2}$ (\cite{Pacard}). The existence of such solution is conjectured in \cite [Remark 2.4]{Pacard}.
We consider the Gierer-Meinhardt system with small inhibitor diffusivity and very small activator diffusivity in a bounded and smooth two-dimensional domain. For any given positive integer $k$ we construct a spike cluster consisting of $k$ boundary spikes which all approach the same nondegenerate local maximum point of the boundary curvature. We show that this spike cluster is linearly stable. The main idea underpinning these stable spike clusters is the following: due to the small inhibitor diffusivity the interaction between spikes is repulsive and the spikes are attracted towards a nondegenerate local maximum point of the boundary curvature. Combining these two effects can lead to an equilibrium of spike positions within the cluster such that the cluster is linearly stable.
We consider the Gierer-Meinhardt system with small inhibitor diffusivity, very small activator diffusivity and a precursor inhomogeneity. For any given positive integer k we construct a spike cluster consisting of $k$ spikes which all approach the same nondegenerate local minimum point of the precursor inhomogeneity. We show that this spike cluster can be linearly stable. In particular, we show the existence of spike clusters for spikes located at the vertices of a polygon with or without centre. Further, the cluster without centre is stable for up to three spikes, whereas the cluster with centre is stable for up to six spikes. The main idea underpinning these stable spike clusters is the following: due to the small inhibitor diffusivity the interaction between spikes is repulsive, and the spikes are attracted towards the local minimum point of the precursor inhomogeneity. Combining these two effects can lead to an equilibrium of spike positions within the cluster such that the cluster is linearly stable.
We prove that finite Morse index solutions to the Allen-Cahn equation in $\R^2$ have {\bf finitely many ends} and {\bf linear energy growth}. The main tool is a {\bf curvature decay estimate} on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second order regularity of clustering interfaces for the singularly perturbed Allen-Cahn equation in $\R^n$. Using an indirect blow-up technique, in the spirit of the classical Colding-Minicozzi theory in minimal surfaces, we show that the {\bf obstruction} to the uniform second order regularity of clustering interfaces in $\R^n$ is associated to the existence of nontrivial entire solutions to a (finite or infinite) {\bf Toda system} in $\R^{n-1}$. For finite Morse index solutions in $\R^2$, we show that this obstruction does not exist by using information on stable solutions of the Toda system.
We consider ground states of two-dimensional Bose-Einstein condensates in a trap with attractive interactions, which can be described equivalently by positive minimizers of the $L^2-$critical constraint Gross-Pitaevskii energy functional. It is known that ground states exist if and only if $a< a^*:= \|w\|_2^2$, where $a$ denotes the interaction strength and $w$ is the unique positive solution of $Δw-w+w^3=0$ in $R^2$. In this paper, we prove the local uniqueness and refined spike profiles of ground states as $a\nearrow a^*$, provided that the trapping potential $h(x)$ is homogeneous and $H(y)=\int_{R^2} h(x+y)w^2(x)dx$ admits a unique and non-degenerate critical point.
From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension $8$, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that Savin's theorem is optimal.
We consider the $2+1$ Toda system \[ \frac{1}{4}Δq_{n}=e^{q_{n-1}-q_{n}}-e^{q_{n}-q_{n+1}}\text{ in }\mathbb{R}^{2},\ n\in\mathbb{Z}. \] It has a traveling wave type solution $\left\{ Q_{n}\right\} $ satisfying $Q_{n+1}(x,y)=Q_{n}(x+\frac{1}{2\sqrt{2}},y)$, and is explicitly given by \[ Q_{n}\left( x,y\right) =\ln\frac{\frac{1}{4}+\left( n-1+2\sqrt{2}x\right) ^{2}+4y^{2}}{\frac{1}{4}+\left( n+2\sqrt{2}x\right) ^{2}+4y^{2}}. \] In this paper we prove that \{$Q_{n}$\} is nondegenerate.
This paper establishes certain existence and classification results for solutions to $SU(n)$ Toda systems with three singular sources at 0, 1, and $\infty$. First, we determine the necessary conditions for such an $SU(n)$ Toda system to be related to an $n$th order hypergeometric equation. Then, we construct solutions for $SU(n)$ Toda systems that satisfy the necessary conditions and also the interlacing conditions from Beukers and Heckman. Finally, for $SU(3)$ Toda systems satisfying the necessary conditions, we classify, under a natural reality assumption, that all the solutions are related to hypergeometric equations. This proof uses the Pohozaev identity.
For all rank two Toda systems with an arbitrary singular source, we use a unified approach to prove: (i) The pair of local masses $(σ_1,σ_2)$ at each blowup point has the expression $$σ_i=2(N_{i1}μ_1+N_{i2}μ_2+N_{i3}),$$ where $N_{ij}\in\mathbb{Z},~i=1,2,~j=1,2,3.$ (ii) Suppose at each vortex point $p_t$, $(α_1^t,α_2^t)$ are integers and $ρ_i\notin 4π\mathbb{N}$, then all the solutions of Toda systems are uniformly bounded. (iii) If the blow up point $q$ is not a vortex point, then $$u^k(x)+2\log|x-x^k|\leq C,$$ where $x^k$ is the local maximum point of $u^k$ near $q$. (iv) If the blow up point $q$ is a vortex point $p_t$ and $α_t^1,α_t^2$ and $1$ are linearly independent over $Q$, then $$u^k(x)+2\log|x-p_t|\leq C.$$ The Harnack type inequalities of (iii) or (iv) is important for studying the bubbling behaves near each blow up point.
In this paper, we compute the Joseph-Lundgren exponent for the quadharmonic Lane-Emden equation, derive a monotonicity formula and classify the finite Morse index solution to the following quadharmonic Lane-Emden equation: \noindent \begin{equation}\nonumber Δ^4 u=|u|^{p-1}u\;\;\;\;\hbox{in}\;\;\;\;\; \R^n. \end{equation} As a byproduct, we also get a monotonicity formula for the quadharmonic maps $ Δ^4 u=0$.
We classify the stable solutions (positive or sign-changing, radial or not) to the following nonlocal Lane-Emden equation: $(-Δ)^s u=|u|^{p-1}u$ in $\mathbb{R}^n$ for $2<s<3$.
We construct Delaunay-type solutions for the fractional Yamabe problem with an isolated singularity $(-Δ)^γw = c_{n, γ} w^{\frac{n+2γ}{n-2γ}}, w>0 \ \mbox{in} \ \mathbb{R}^n \backslash \{0\}$ We follow a variational approach, in which the key is the computation of the fractional Laplacian in polar coordinates.
We derive a monotonicity formula and classify finite Morse index solutions (positive or sign-changing, radial or not) to the following triharmonic Lane-Emden equation: \begin{equation}\nonumber (-Δ)^3 u=|u|^{p-1}u \hbox{ in } \mathbb{R}^n, \end{equation} where $p$ is below the Joseph-Lundgren exponent. As a byproduct we also obtain a new monotonicity formula for the triharmonic maps.
The note is aimed at giving a complete characterization of the following equation: $$\displaystyle p\frac{Γ(\frac{n}{2}-\frac{s}{p-1})Γ(s+\frac{s}{p-1})}{Γ(\frac{s}{p-1})Γ(\frac{n-2s}{2}-\frac{s}{p-1})} =\frac{Γ(\frac{n+2s}{4})^2}{Γ(\frac{n-2s}{4})^2}.$$ The method is based on some key transformation and the properties of the Gamma function. Applications to fractional nonlinear Lane-Emden equations will be given.
We prove the existence of global minimizers of Allen-Cahn equation in dimensions $8$ and above. More precisely, given any strictly area-minimizing Lawson's cones, there are global minimizers whose nodal sets are asymptotic to the cones. As a consequence of Jerison-Monneau's program we establish the existence of many counter-examples to the De Giorgi's conjecture different from the Bombierie-De Giorgi-Giusti minimal graph.
We consider the $SU(n+1)$ Toda system $$(S_λ) \quad \left\{ \begin{aligned} & Δu_1 + 2λe^{u_1} - λe^{u_2}- \dots - λe^{u_k} = 0\quad \hbox{in}\ Ω,\\ & Δu_2 - λe^{u_1} + 2λe^{u_2} - \dots - λe^{u_k}=0\quad \hbox{in}\ Ω,\\ &\vdots \hskip3truecm \ddots \hskip2truecm \vdots\\ & Δu_k -λe^{u_1}-λe^{u_2}- \dots+2λe^{u_k}=0\quad \hbox{in}\ Ω, &u_1 = u_2 = \dots = u_k =0 \quad \hbox{on}\ \partialΩ.\\ \end{aligned}\right. $$ If $0\inΩ$ and $Ω$ is symmetric with respect to the origin, we construct a family of solutions $({u_1}_λ,\dots,{u_k}_λ)$ to $(S_λ)$ such that the $i-$th component ${u_i}_λ$ blows-up at the origin with a mass $2^{i+1}π$ as $λ$ goes to zero.
Starting from a bound state (positive or sign-changing) solution to $$ -Δω_m =|ω_m|^{p-1} ω_m -ω_m \ \ \mbox{in}\ \R^n, \ ω_m \in H^2 (\R^n)$$ and solutions to the Helmholtz equation $$ Δu_0 + λu_0=0 \ \ \mbox{in} \ \R^n, \ λ>0, $$ we build new Dancer's type entire solutions to the nonlinear scalar equation $$ -Δu =|u|^{p-1} u-u \ \ \mbox{in} \ \R^{m+n}. $$